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In probability theory and statistics, the cumulative distribution function of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} , evaluated at x {\displaystyle x} , is the probability that X {\displaystyle X} will take a value less than or equal to x {\displaystyle x} .
Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an upwards continuous monotonic increasing cumulative distribution function F : R → {\displaystyle F:\mathbb {R} \rightarrow } satisfying lim x → − ∞ F = 0 {\displaystyle \lim _{x\rightarrow -\infty }F=0} and lim x → ∞ F = 1 {\displaystyle \lim _{x\rightarrow \infty }F=1} .
In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x {\displaystyle x} . Cumulative distribution functions are also used to specify the distribution of multivariate random variables.